If E is an elliptic curve over Q
and V
is an Artin representation of Q, a natural
equivariant refinement of the Birch and Swinnerton-Dyer conjecture relates the order
of vanishing of the Hasse-Weil-Artin L-series L(E,V,s) at s=1
to the multiplicity with which V appears in the Mordell-Weil group
of E over the algebraic closure of Q,
viewed as a Galois representation.
In particular, it implies that V does not appear when L(E,V,1) is non-zero.
I will outline the ideas entering into the proof of this last statement for certain
irreducible Artin representations V of dimensions 1, 2 and 4.
The one-dimensional setting is a landmark result of Kato, and the
two and four-dimensional settings are the object of ongoing joint projects
with Bertolini-Rotger and Rotger respectively, building on
Kato's fundamental insights.